Let's go through this hypothesis test for the population slope $\beta_1$ of the regression line.

The regression equation is given as:

$y = 7.971 + 1.97x$

with the standard error of the slope being approximately $1.64$.

Hypothesis Test Steps

(a) State the Null Hypothesis ($H_0$) and Alternative Hypothesis ($H_1$):

• $H_0: \beta_1 = 0$ (There is no linear relationship between the 1980 and 1999 per capita income.)
• $H_1: \beta_1 \neq 0$ (There is a significant linear relationship between the 1980 and 1999 per capita income.)

(b) Determine the Type of Test Statistic:

This is a t-test because we are testing the slope of a linear regression model.

Degrees of freedom (df) can be calculated using:

$df = n - 2$

where $n = 50$ (the number of U.S. states). So,

$df = 50 - 2 = 48$

(c) Find the Value of the Test Statistic:

The test statistic $t$ for the slope is calculated as:

$t = \frac{\hat{\beta}_1 - 0}{SE_{\hat{\beta}_1}}$

where:

• $\hat{\beta}_1 = 1.97$ (the estimated slope),
• $SE_{\hat{\beta}_1} = 1.64$ (the standard error of the slope).

Substituting the values:

$t = \frac{1.97 - 0}{1.64} = 1.201$

(d) Find the Two Critical Values:

For a two-tailed test at the $0.10$ significance level, we look up the critical values for $df = 48$ and a significance level of $0.10$ (i.e., $0.05$ in each tail). Using a t-distribution table or calculator:

$t_{0.05, 48} \approx \pm 1.677$

So, the critical values are approximately $-1.677$ and $+1.677$.

(e) Conclusion:

Since the test statistic $t = 1.201$ is within the range $-1.677 \leq t \leq 1.677$, we fail to reject the null hypothesis $H_0$. This means that, at the $0.10$ significance level, there is insufficient evidence to conclude that there is a significant linear relationship between the 1980 and 1999 per capita income for U.S. states.

Summary:

• Test statistic: $t = 1.201$
• Critical values: $\pm 1.677$
• Conclusion: No significant linear relationship at the 0.10 level.

Would you like further details on any part of this process or the formulas involved?

Here are 5 related questions for further exploration:

1. How would the conclusion change if the significance level was 0.05 instead of 0.10?
2. What impact does a higher or lower standard error have on the test statistic?
3. How do you calculate the confidence interval for the slope $\beta_1$?
4. How does increasing the sample size affect the degrees of freedom and critical values?
5. Can you explain how residuals are used to check regression assumptions?

Tip: When performing hypothesis tests for regression, always verify that the regression assumptions (linearity, normality, homoscedasticity, and independence) hold.